Traveling-wave parametric amplifiers, first proposed in the late 1950s, rely on the use of a low-loss nonlinear material. Although originally envisioned as electrical amplifiers operating at radio or microwave frequencies, the traveling-wave parametric amplifier concept was later adapted for use at optical frequencies and versions using optical fibers have now been successfully demonstrated. Relevant descriptions of those technologies include A. L. Cullen. A travelling-wave parametric amplifier. Nature, 181:332, 1958; P. K. Tien. Parametric Amplification and Frequency Mixing in Propagating Circuits. Journal of Applied Physics, 29:1347-1357, September 1958; A. L. Cullen. Theory of the travelling-wave parametric amplifier. Proc. IEE—Part B, 107(32):101-107, 1960; J. Hansryd and P. A. Andrekson. Broad-band continuous-wave-pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency. IEEE Photonics Technology Letters, 13:194-196, March 2001; and J Hansryd, P A Andrekson, M Westlund, J Li, and PO Hedekvist. Fiber-based optical parametric amplifiers and their applications. IEEE J. Sel. Top. Quantum Electron., 8(3):506-520, 2002.
The use of the nonlinear response of a physical system to obtain oscillation or amplification has a long history. See for example, W. W. Mumford. Some notes on the history of parametric transducers. Proc. IRE, 48:848-853, May 1960. In the 1950s, the development of low-loss diodes with voltage-variable capacitance—varactors—stimulated a resurgence of interest in such effects and allowed the construction of low noise amplifiers operating in the microwave to millimeter-wave bands. See for example, L. S. Nergaard. Nonlinear-capacitance amplifiers. RCA Review, pages 3-17, March 1959. These devices are known as parametric amplifiers, because one of the circuit parameters (the capacitance in this case) is made to vary periodically through the application of a strong “pump” voltage waveform at frequency fP. If a weak signal waveform at frequency fS is also applied to the circuit, under proper conditions the time-varying capacitance may transfer power from the pump frequency to the signal frequency, thereby achieving amplification. The pump frequency may either be around the signal frequency or around twice the signal frequency depending on the type of nonlinearity. Parametric amplification also involves generation of a third frequency, the “idler” f1. The three frequencies are related by fS+f1=fP or fS+f1=2fP, depending on the type of nonlinearity. The special case fS=f1 is called the “degenerate mode”; otherwise the amplifier is said to be “non-degenerate”.
Because the amplification mechanism relies on a nonlinear reactance, the device need not produce thermal noise or shot noise, and by the early 1960s it was understood that in principle, the noise performance is limited only by quantum mechanics. See, for example, W. H. Louisell, A. Yariv, and A. E. Siegman. Quantum Fluctuations and Noise in Parametric Processes. I. Physical Review, 124:1646-1654, December 1961. In practice, varactor diodes are not ideal and do produce some excess noise, but nevertheless excellent noise performance can be obtained. Due to their low noise, diode-based parametric amplifiers played a key role in the development of satellite communication and radio astronomy before they were replaced by cryogenic transistor amplifiers. Relevant descriptions include S. Weinreb, M. Balister, S. Maas, and P. J. Napier, Multiband low-noise receivers for a very large array. IEEE Transactions on Microwave Theory Techniques, 25:243-248, April 1977; Sander Weinreb, Tests of cooled 5 GHz parametric and GASFET amplifiers. Electronics Division Internal Report 203, National Radio Astronomy Observatory, Charlottesville, Va., 1980; M. E. Hines. The Virtues of Nonlinearity—Detection, Frequency Conversion, Parametric Amplification and Harmonic Generation. IEEE Transactions on Microwave Theory Techniques, 32:1097-1104, September 1984; and J. J. Whelehan, Low-noise amplifiers-then and now. IEEE Transactions on Microwave Theory Techniques, 50:806-813, March 2002. Today, cryogenic amplifiers using both high electron mobility transistors (HEMT) and bipolar silicon-germanium (SiGe) transistors offer excellent performance in the microwave band, with noise temperatures dropping below 1 K/GHz. However, this noise performance is still a factor of ˜20 above the quantum mechanical limit hν/kB≈48 mK/GHz, and is inadequate for some demanding applications. Indeed, there is currently great interest in the topic of quantum-limited amplification and measurement, particularly at microwave frequencies. See, for example, A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf. Introduction to quantum noise, measurement, and amplification. Reviews of Modern Physics, 82:1155-1208, April 2010.
Traveling-Wave Parametric Amplifiers
In order to deal with the capacitive reactance of the varactor diodes, early parametric amplifiers used single diodes in tuned circuits and consequently had fairly narrow bandwidths. The desire to increase the bandwidth led to the idea of the traveling-wave parametric amplifier, patented by Tien (U.S. Pat. No. 3,012,203), in which the nonlinear devices were placed periodically along the length of a distributed structure. If properly designed, interactions between the pump and signal waves propagating along this structure could lead to amplification of the signal. However, Landauer (R. Landauer, Parametric Amplification along Nonlinear Transmission Lines. Journal of Applied Physics, 31:479-484, March 1960, and R. Landauer, Shock waves in nonlinear transmission lines and their effect on parametric amplification. IBM J. Res. Dev., 4(4):391-401, 1960.) realized that the conditions required for parametric amplification in a traveling-wave structure would also, in general, lead to the development of shock fronts in the pump waveform. Landauer argued that traveling-wave amplifiers would be crippled by this effect: shock wave formation would prevent the amplifiers from achieving the desired combination of high gain with low noise.
Judging from the literature, Landauer's work appears to have substantially dampened the enthusiasm for traveling-wave parametric amplifiers operating at microwave frequencies in the subsequent decades. However, by the early 1960s the invention of the laser had raised the intriguing possibility of achieving parametric amplification and oscillation at optical wavelengths, and the traveling-wave theory developed by Tien for electrical amplifiers was adapted for the optical case. See, for example, R. H. Kingston, Parametric amplification and oscillation at optical frequencies. Proc. IRE, 50:472, 1962; and N. M. Kroll, Parametric Amplification in Spatially Extended Media and Application to the Design of Tuneable Oscillators at Optical Frequencies. Physical Review, 127:1207-1211, August 1962. The use of an optical fiber to confine the light is very helpful for maximizing the power density and field strength for a fixed optical power, and this enhances the nonlinear effects required for parametric amplification. The idea of using an optical fiber for optical traveling wave parametric devices was described in a 1967 patent awarded to Garwin, Hardy, and Landauer (U.S. Pat. No. 3,297,875). Apparently, at this point Landauer understood that the significant dispersion (the wavelength-dependent phase velocity) of optical fibers would render moot his previous shock-wave objections to traveling-wave amplifiers. However, optical fiber parametric amplifiers did not receive much attention for decades, until 2001 when Hansryd and Andrekson demonstrated experimentally that, by using the dispersive characteristics of the fiber to advantage, very high gain was in fact possible in a fiber parametric amplifier.
Superconducting Parametric Amplifiers
The fact that superconductors exhibit nonlinear behavior at microwave frequencies was already known in 1950. See, for example, A. B. Pippard, Field variation of the superconducting penetration depth. Proc. Roy. Soc. A, 203(1073):210-223, 1950; and A. B. Pippard. An experimental and theoretical study of the relation between magnetic field and current in a superconductor. Proc. Roy. Soc. A, 216(1127):547-568, 1953. The development of a microscopic theory of superconductivity by Bardeen, Cooper and Schrieffer in 1957 quickly led to the Mattis-Bardeen theory of the linear electrodynamic response. However, for large currents approaching the critical value, the response is expected to become nonlinear, as discussed theoretically by Parmenter in 1962. In essence, the kinetic inductance of the superconductor is expected to increase at high currents, with the change in inductance varying as the square of the current. Connell's 1963 experimental study of this effect in Indium alloys led him to propose a parametric amplifier using thin-film superconducting transmission lines. However, Connell focused his attention on standing-wave transmission-line resonators, as suggested and patented by Landauer (U.S. Pat. No. 3,111,628); presumably the shock-wave issue deterred serious consideration of the traveling-wave version. The first experimental demonstration of parametric effects using superconducting films (tin) was given by Clorfeine. See, for example, A. S. Clorfeine, Microwave amplification with superconductors. Proc. IEEE, pages 844-845, 1964; and A. S. Clorfeine, Nonlinear Reactance and Frequency Conversion in Superconducting Films at Millimeter Wavelengths. Applied Physics Letters, 4:131-132, April 1964. In this experiment, the extremely large dielectric constant of rutile (TiO2) was harnessed in a waveguide-coupled dielectric resonator in order to couple to the very low surface impedance of the superconducting film in order to obtain 11 dB gain at 6 GHz with a gain-bandwidth product of around 1 MHz. This narrow bandwidth meant that the device had little practical value beyond a basic demonstration, and while the topic continued to receive theoretical attention on occasion, the focus of experimental work shifted to other more promising directions.
Josephson Parametric Amplifiers
In 1967, Zimmer reported on experiments similar to those of Clorfeine. He conjectured that Clorfeine's results were actually due to the nonlinearity of Josephson currents in naturally-occurring barriers (or weak links) in the very thin films being used, rather than the nonlinear behavior of the superconducting film itself. He then repeated the experiment using samples with intentionally fabricated Josephson junctions and found that the junction samples reliably showed parametric amplification. This triggered strong interest in the development of Josephson-junction parametric amplifiers, and by the late 1980s improved junction fabrication techniques allowed more sophisticated devices to be produced reliably, and narrowband microwave amplifiers with excellent noise performance approaching the quantum limit were demonstrated.
Traveling-Wave Josephson Parametric Amplifiers
In an effort to improve the bandwidth, the traveling-wave Josephson parametric amplifier was proposed 25 years ago (M. Sweeny and R. Mahler, A travelling-wave parametric amplifier utilizing Josephson junctions. IEEE Trans. Magn., 21:654, March 1985), and is shown in FIG. 1A. Amplifiers of this type have in fact been demonstrated and have a gain-bandwidth product comparable to or better than any superconducting parametric amplifier to date. Such devices have recently received renewed attention. The traveling-wave amplifier uses a series array of Josephson junctions that are distributed along a coplanar waveguide transmission line. The junctions are in their superconducting state and can carry a Josephson supercurrent. The associated “Josephson inductance” effectively contributes a series inductance  per unit length to the transmission line, and this inductance increases with the current I until the junction critical current is reached, at which point the junctions switch into their resistive state and the line suddenly becomes very lossy. Below the critical current, the line remains low-loss, but the phase velocity =1/√{square root over (C)} of a wave decreases as the wave amplitude increases. This nonlinear behavior is equivalent to the Kerr effect in optical materials, in which the refractive index is intensity dependent.
The operation of this device can be understood in a straightforward way. Imagine a strong pump wave at frequency fP traveling down the transmission line, emerging at the output end. If we change the pump amplitude, the phase of the wave at the output will change due to the amplitude-dependent phase velocity of the line. Now, imagine that the pump amplitude is kept constant, but a small “signal” wave fS is added at a nearby frequency. The sum of the pump and signal waves produces an amplitude envelope that varies sinusoidally at the difference or “beat” frequency |fS−fP|, and therefore the wave that emerges at the output has a phase φ that also varies at this difference frequency. In other words, the nonlinear transmission line transfers amplitude modulation into phase modulation. If the nonlinearity is sufficiently strong, this effect can produce gain at the signal frequency. In addition, phase modulation of the pump implies the existence of an “idler” frequency at the output, whose frequency is located at the “mirror image” of the signal frequency with respect to the pump, f1=2fP−fS.
FIG. 1A is a schematic diagram of a prior art traveling-wave Josephson parametric amplifier proposed by Sweeny and Mahler in 1985 (M. Sweeny and R. Mahler, cited above). FIG. 1B is a plan diagram of the traveling-wave Josephson parametric amplifier shown in FIG. 1A. FIG. 1C is a graph showing measured gain (in dB) vs. signal frequency offset (|fS−fP|) of a 20 GHz Josephson parametric amplifier of the type shown in FIG. 1A, as demonstrated by Yurke et al. (B. Yurke, M. L. Roukes, R. Movshovich, and A. N. Pargellis. A low-noise series-array Josephson junction parametric amplifier. Appl. Phys. Lett., 69:3078-3080, November 1996). The gain-bandwidth product of this device is 500 MHz. The input and output impedance match for this particular device was poor, so it was likely operating in a partial standing-wave mode. Another example is described by E. M. Levenson-Falk et al, arXiv:1101.4672v1 [cond-mat.supr-con] 24 Jan. 2011. The amplifier described in that paper has sqrt(G) of about 10 over 40 MHz, so it appears to have a gain bandwidth product of 400 MHz.
Also known in the prior art is Lentz, U.S. Pat. No. 2,962,681, issued Nov. 29, 1960, which is said to disclose superconductor circuits fabricated in the form of transmission lines and including superconductor gate conductors which are selectively driven resistive to control the current in the circuits, wherein the resistance of the gate conductors is so related to the characteristic impedance of the transmission lines that optimum switching speeds are obtained.
Also known in the prior art is Tien, U.S. Pat. No. 3,012,203, issued Dec. 5, 1961, which is said to disclose parametric amplification by controlled reactance variation into the domain of traveling waves. It provides an extended, traveling wave-supporting structure, proportioned to support traveling waves of the frequencies of interest and preferably, also, to suppress waves of frequencies not of interest. Among the waves of interest is a wave of reactance variation launched by the source of pumping energy. The others are the wave to be amplified and at least one auxiliary wave. The structure is proportioned to cause the waves of interest to travel with the phase speeds and group speeds that are most advantageous in promoting the gain-producing interaction among them; in particular with the phase speeds that satisfy a certain relation among the phase constants.
Also known in the prior art is Landauer, U.S. Pat. No. 3,111,628, issued Nov. 19, 1963, which is said to disclose improved parametric circuits of the distributed constant type which utilize the phenomenon of superconductivity. Briefly, this invention includes a non-linear superconductive transmission line wherein the non-linearity of the transmission line is a function of the penetration depth of magnetic fields into the conductors of the line.
Also known in the prior art is Garwin et al., U.S. Pat. No. 3,297,875, issued Jan. 10, 1967, which is said to disclose a traveling wave parametric device operable at optical frequencies that is provided by applying to a non-linearly polarizable medium one or more optical waves, the medium being capable of supporting a plurality of optical waves which are frequency related to the applied wave or waves and for which the wave velocities are matched to the supporting medium, this latter condition expressing the requirement that this plurality of waves interferes constructively from point to point as they commonly propagate through the supporting medium.
There is a need for improved traveling wave parametric amplifiers.